ISSN 0015-4628, Fluid Dynamics, 2013, Vol. 48, No. 2, pp. 239–250. © Pleiades Publishing, Ltd., 2013.
Original Russian Text © A.N. Volkov, R.E. Johnson, O.J. Tucker, 2013, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i
Gaza, 2013, Vol. 48, No. 2, pp. 112–125.
Expansion of Monatomic and Diatomic Gases
from a Spherical Source into a Vacuum
in a Gravitational Field
A. N. Volkov, R. E. Johnson, and O. J. Tucker
Received June 17, 2011
Abstract—Steady monatomic and diatomic gas flows from a spherical source into a vacuum in a gravitational field are studied using direct statistical simulation. The qualitative gravitation effect on the flow
is shown to be independent of the intermolecular collision model. Three characteristic Jeans parameter
ranges can always be distinguished, namely, the subcritical range, on which the flow in a weak gravitational field is similar with the outflow in the absence of gravitation, the supercritical range, on which the
outflow velocity remains small even at large distances from the source, and a narrow transitional range
between the two former ranges. The presence of internal degrees of freedom of gas molecules displaces
the transitional range toward the greater values of the Jeans parameter and leads to an increase in the
outflow velocity and the gas temperature; however, in the initial region the latter effect is expressed only
slightly. The normalized escape flow is a nonmonotonic function of both the Jeans parameter and the
Knudsen number and is different for monatomic and diatomic gases within 50% on the parameter range
considered.
Keywords: spherical source, expansion into a vacuum, gravitation, diatomic gas, thermal escape, escape flux.
DOI: 10.1134/S0015462813020117
The problem of gas expansion into a vacuum in a gravitational field is of interest in certain problems of
astrophysics and physics of the atmosphere, in particular, for calculating the structures of upper planetary
atmospheres and their evolution on geological time scales [1]. The upper atomosphere, where the molecular
free path is greater than the vertical density variation scale, is called exosphere. In certain cases [2], for
example for the Pluto’s atmosphere, the interaction between the exosphere molecules and the solar wind
plays a secondary role. Then the molecules that in the free-molecular flow region have a kinetic energy
sufficient for overcoming the gravity field of the planet escape from the atmosphere. At the same time, an
upward-directed thermal flux formed in the atmosphere ensures the energy supply to the exosphere, which
would be necessary for maintaining a constant escape flux, that is, the number of molecules leaving the
exosphere in hyperbolic trajectories per unit time. Precisely this process called thermal escape [1, 2] is the
subject of this study.
Despite the fact that the exosphere includes regions of transitional and free-molecular flows, the kinetic
models of exospheric flows are chiefly applied only for calculating the thermal escape of molecules of
“light” admixtures diffusing in the equilibrium field of the main “heavy” atmospheric fractions [3] and
for calculating nonthermal escape mechanisms (see reference in [1]). In modeling the flow of the main
atmospheric fractions it is the models based on the continual equations of steady spherically-symmetric
flow of a heat-conducting gas in a gravitational field [4–6] that are employed. The use of these models in
the problems of the atmosphere dynamics dates back to [7, 8], where a similar model was developed for the
problems of solar wind dynamics.
The kinetic problem of thermal escape in a spherically-symmetric atmosphere is similar with the problem
of gas outflow from a spherical source into a vacuum in the absence of gravitation. The latter problem
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VOLKOV et al.
was theoretically studied by means of solving the kinetic Bhatnagar–Gross–Krook (BGK) equation in the
hypersonic approximation [9, 10], by numerically solving the BGK equation and its generalizations [11–13],
and by direct statistical simulation (DSS) [14–16]; it was also experimentally studied [17]. The theoretical
studies showed that at low Knudsen numbers at the source surface the flow is nonequilibrium in the Knudsen
layer [15, 16] and in the far field [9, 10, 13], whereas in the intermediate region it can be described by the
Navier–Stokes equations with the boundary conditions obtained in the kinetic calculations [15, 16, 18].
The kinetic calculations of the escape flux of the main atmospheric fractions in a gravitational field are
often based on an approximate theory which assumes that the flow is equilibrium on the nominal boundary of
the exosphere and is free-molecular above it [1]. The effect of the kinetics on the exosphere flow structure
was apparently first considered in [19] on the basis of the numerical solution of the BGK equation, In
particular, in [19] it was shown that in the presence of a gravitational field intermolecular collisions lead to
the appearance of a considerable fraction of molecules which move along elliptical trajectories; however,
the calculations were restricted to the case of fairly high Knudsen numbers. The kinetic calculations of flows
of a monatomic solid-sphere gas in a gravitational field on a wide Knudsen number range [20–22] showed
that taking the nonequilibrium into account leads to both a qualitative difference in the exosphere structure
and a considerable change in the escape flux, as compared with the results based on gasdynamic models.
In this study the purpose of the calculations is to establish the effects of the intermolecular collision
model and their internal degrees of freedom on the laws of gas expansion in a gravitational field and the
escape flux. The flow in a single-component atmosphere is calculated using direct statistical simulation.
The formulation of the problem is analogous to that of free gas expansion into a vacuum [13, 15, 16], while
the main differences are due to the presence of the gravitational field.
1. KINETIC MODEL OF THERMAL ESCAPE
We will consider one-dimensional steady single-component-gas outflow from a spherical surface, R0 in
radius. It is assumed that any gas molecule with a mass m is exposed to the force FG = −GMm/r2 acting
from a spherically-symmetric field produced by a body of mass M, whose center coincides with the source
center (here, G is the gravitational constant and r is the distance to the source center). We will use three
models for describing the intermolecular interaction.
(1) The model of molecules as hard spheres, d in diameter, (HS model) with the differential scattering
cross-section σ = d 2 /4.
(2) The model of pseudo-Maxwellian molecules (PM model) in which the collisions between molecules
of a monatomic gas are described by the model of hard spheres of variable diameter [23] with a differential
scattering cross-section σ = σ0 g0 /g, where g is the relative velocity of colliding molecules and σ0 g0 is a
model parameter.
(3) The model of diatomic gas with internal degrees of freedom (PMLB model) [23] in which the scattering cross-sections are the same as for the pseudo-Maxwellian molecules, while the energy exchange
between the translational and internal degrees of freedom of molecules in collisions is described by the
Larsen–Borgnakke model [24]. In all the calculations (except those whose results are presented as curves III
in Fig. 1a) the number of the internal degrees of freedom ζ is taken to be two. This particular case can be
important in calculating the atmospheres of certain planets, for example, Titan [5, 20] and Pluto [6, 25], in
which the main component is nitrogen N2 whose molecules at a characteristic temperature of about 100 K
have two excited rotational degrees of freedom. In the calculations the probability of nonelastic collisions
is taken to be unity.
In what follows the formulation of the problem is considered only for the PMLB model. For the HS
and PM models the formulations of the problem can be derived from that for the PMLB model by means of
certain simplifications connected with the absence of internal degrees of freedom from the gas molecules.
The gas flow is described in terms of the molecular distribution function f (r, v∥ , v⊥ , εi , t) in the radial
v∥ and transverse v⊥ velocity components and the internal energy εi (t is time). The distribution function
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Fig. 1. (a) Dependences of the local Mach number M (I) and the dimensionless velocity u/C0 (II) and temperature T /T0
(III) on the distance r/R0 : (I–III), this study; (1) [13]; (2) [15]; and (3) [16]; (I) √
and (2) HS model, Kn0 = 10−4 ; (II) and
−3
−3
(1) PM, Kn0 = 10 ; and (III) and (3) PMLB with ζ = 3 and Kn0 = 10 ; C0 = 2kT0 /m; (b) Dependence of M on r/R0
at different λ0 : PMLB model, Kn0 = 10−3 , λ0 = 0, 1, 2, 2.7, 4, 10, and 15, curves (1–7).
satisfies the Boltzmann equation [11, 21, 26]
( 2
)
v∥ v⊥ ∂ f
v⊥
FG ∂ f
∂f
∂f
+ v∥
+
+
−
= IB ( f , f ),
∂t
∂r
r
m ∂ v∥
r ∂ v⊥
(1.1)
where IB ( f , f ) is the Boltzmann collision integral which is not written down here, since it is not used in
what follows; however, for the HS model it is presented in [27]. The left side of Eq. (1.1) governs the
variation of f (r, v∥ , v⊥ , εi , t) due to the motion of molecules in their orbital planes.
It is assumed that on the source surface the molecules with v∥ > 0 obey the Maxwell–Boltzmann distribution at a given concentration n0 , zero macroscopic velocity u0 , and a temperature T0 [23, 28]
v∥ > 0 : f (R0 , v∥ , v⊥ , εi , t) = f0t (v∥ , v⊥ ) f0i (εi ),
(1.2)
(
)
m(v2∥ + v2⊥ )
n0
exp −
,
f0t (v∥ , v⊥ ) =
2kT0
(2π kT0 /m)3/2
(1.3)
f0i (εi ) =
(
)
ζ /2 − 1
εi
εi
exp
−
.
kT0
(kT0 )ζ /2 Γ(ζ /2)
(1.4)
Here, Γ(x) is the gamma function, while the molecules that return onto the source surface are absorbed by
it. Thus, the boundary conditions (1.2)–(1.4) do not impose explicit restrictions on the distribution function
of the molecules that return onto the source surface, while the actual values of n, u, and T on the surface
differ from n0 , u0 = 0, and T0 . The conditions analogous to (1.2)–(1.4) were used in [13, 15, 16], while in
[11] it was assumed that on the source surface u0 is equal to the local speed of sound.
It is assumed that the exit boundary r = R1 is located fairly far away from the source surface, so that the
flow on this boundary can be approximately assumed to be free-molecular. Then on the exit boundary the
molecular distribution function must satisfy the boundary condition [21]
⎧
⎨ f (R1 , −v∥ , v⊥ , εi , t), v < ve (R1 ),
v∥ = 0 : f (R1 , v∥ , v⊥ , εi , t) =
⎩
0,
v ≥ ve (R1 ),
√
where
√ ve (r) = 2GM/r is the escape velocity of a molecule at a distance of r from the source center and
v = v2∥ + v2⊥ . On the upper boundary of the computation domain the molecules with v∥ > 0 and v > ve (r)
move without collisions along hyperbolic trajectories, thus forming the escape flux.
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In the steady flow the distribution function f (r, v∥ , v⊥ , εi ) can be calculated using the stabilization
method as a solution realized, as t → ∞. In this study the calculations using the stabilization method were
made under the condition that at the initial moment t = 0 the computation domain is empty but then it is
gradually filled with gas molecules at the expense of “evaporation” governed by Eqs. (1.2)–(1.4).
The macroscopic gas parameters can be represented in the form:
+∞
∫ ∫∞ ∫∞
⟨Ψ⟩(r) = 2π
Ψ(r, v∥ , v⊥ , εi ) f (r, v∥ , v⊥ , εi )v⊥ d εi dv⊥ dv∥ ,
−∞ 0 0
where Ψ(r, v∥ , v⊥ , εi ) are different physical quantities characterizing individual molecules. The main
macroscopic parameters considered below are the concentration n (Ψ = 1), the gas velocity u (Ψ = v∥ /n),
the radial T∥ (Ψ = m(v∥ − u)2 /(nk)) and transverse T⊥ (Ψ = mv2⊥ /(2nk)) translational temperatures, the
temperatures of the translational Tt = (T∥ + 2T⊥ )/3 and internal Ti (Ψ = εi /(nk)) degrees of freedom, the
√
total temperature T = (T∥ + 2T⊥ + ζ Ti )/(3 + ζ ) and the local Mach number M = u/ γ kT /m, where
k is the Boltzmann constant and γ = (5 + ζ )/(3 + ζ ) is the Poisson constant. Here, the definition of
the Mach number is based on the formula for the speed of sound which has meaning only in the nearequilibrium flow region. Nevertheless, the quantity M can be conveniently used in analyzing the entire
flow structure, as a parameter characterizing the ratio of the energies of the ordered and chaotic motions
of gas molecules [9, 13]. The escape flux Φ can also be represented in the form Φ = ⟨Ψ⟩(R1 ), where
Ψ = 4π r2 v∥ H(v∥ )H(v − ve (r)) and H(x) is the Heaviside function, H(x) = 1 for x > 0 and H(x) = 0 for
x ≤ 0. Along with the actual escape flow Φ, we will also consider the local escape flux ϕ (r) = ⟨Ψ⟩(r)
calculated for a given parameter distribution in the atmosphere provided that outside the sphere, r in radius,
the molecular motion is collisionless [21].
For the models under consideration the flow in the dimensionless form is determined by two similarity
criteria. For these criteria the Knudsen number Kn0 = l0 /R0 and the Jeans parameter λ0 = R0 /H0 (ratio of
the gravitational energy of a molecule to its characteristic energy of the chaotic motion) on the source surface
can
√ path under equilibrium conditions on the source surface (l0 =
√ be taken; here, l0 is the molecular free
( 2π d 2 n0 )−1 for the HS model and l0 = 8kT0 /(π m)/(4πσ0 g0 n0 ) for the PM and PMLB models [23]) and
H0 = kT0 R20 /(GMm) is the vertical density-variation scale [1]. In this study, we consider the λ0 = 0–15 range
which is of interest for the atmospheres of comets and other small bodies [18, 29], Pluto [6, 25], and the
earlier Earth [4], as well as for chiefly hydrogen atmospheres of certain exoplanets. The Knudsen number
range under consideration, Kn0 ≥ 10−3 , is sufficient for describing the rarefied region of an atmosphere,
where solar radiation absorption is often slight.
The calculations are carried out using the DSS method based on the Bird scheme [23]. The DSS method
may be considered as a numerical method for estimating functionals of the solution of the Boltzmann equation [30, 31], whose accuracy depends on the values of the main numerical parameters which include the
dimensions of the computation grid cells, the time step, and the number of particles in a cell. In this study,
we use nonuniform grids with the cell size increasing with distance from the source [27]. The numerical
parameters of the DSS method were chosen in accordance with the recommendations given in [23]. In particular, the computational grid cell dimensions and the time step were not greater than 30% of the local
molecular free path and time, while the particle number within one cell was varied by several decimal orders but was not less than 30. The majority of the results was obtained at R1 /R0 = 40 but some calculations
were made at R1 /R0 = 6, 10, and 100. The time step number necessary for obtaining a steady-state flow
depended on Kn0 , λ0 , and R1 /R0 and ranged from about 106 to about 108 . The calculation accuracy was estimated in a set of calculations in which the numerical method parameters were varied minimum by several
times. The error in calculating the escape flux was not greater than 7%, the maximum error being realized
in a narrow interval of λ0 which approximately corresponded to the limit of isentropic outflow existence
(λ0 = 2–4 for the HS model), whereas outside this interval the error was several times smaller. On this λ0
interval at small Kn0 the effect of the outer boundary can propagate far upstream [21]. For this reason, the
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flow structures calculated at 2 ≤ λ0 ≤ 4 and shown in Figs. 1 to 4 and 7 for r/R0 ≤ 40 were obtained at
R1 /R0 = 100. Outside the above-mentioned interval the outer boundary has only a slight effect on the flow
structure throughout the entire computation domain, while the escape flux value is almost independent of
R1 /R0 at R1 /R0 ≥ 10–40. Thus, in [21] it was shown that at λ0 = 10 and Kn0 = 0.01 the variation of R1 /R0
from 5 to 100 results in an escape flux variation within 3%.
2. FLOW STRUCTURE AND NONEQUILIBRIUM EFFECTS
The flow structures obtained in this study at λ0 = 0 (Fig. 1a) are in good (with an accuracy of 1%)
agreement with the known results obtained using the DSS method [15, 16] and moderately differ from the
direct numerical solution of the BGK equation [13] (the difference is less than 6% for u/C0 and less than
9% for T /T0 , see Fig. 1a). At λ0 = 0 and Kn0 ≪ 1 the outflow is characterized by hypersonic values of the
limiting Mach number which monotonically increase with decreasing Kn0 .
In [21] it was found that in the problem of spherical outflow of monatomic hard-sphere gas into a vacuum
in a gravitational field three flow regimes can exist depending on λ0 . On the subcritical Jeans parameter
range (λ0 ≤ λc1 ≈ 2.1) the outflow is similar with that in the absence of the gravitational field. At Kn0 ≪ 1
it is characterized by hypersonic limiting Mach numbers. In the supersonic near-source region this outflow
can be described by the isentropic model. On the supercritical Jeans parameter range (λ0 ≥ λc2 ≈ 3–4) the
flow is maintained at the expense of a thermal flux directed from the source and is characterized by very low
Mach numbers up to the lower boundary of the exosphere. On a narrow transitional range (λc1 ≤ λ0 ≤ λc2 )
the flow rapidly goes over from subcritical into supercritical.
The results of the calculations show that this qualitative pattern remains without changes irrespective of
the chosen molecular collision model. In particular, in a diatomic gas an increase in λ0 from 0 to about 3
results in a gradual decrease in the gradient of the local Mach number M and, correspondingly, the limiting
Mach number (curves (1–4) in Fig. 1b). On the ∼ 3 ≤ λ0 ≤∼ 4–5 range (curve 5 in Fig. 1b) the Mach
number steeply reduces throughout the entire flow region, including the source surface. This is due to the
fact that the limit of isentropic outflow existence has been attainted [21, 22]: for λ0 >∼ 3 the initial enthalpy
of fluid particles on the source surface is insufficient to escape from the gravitational well of the source. As a
result, with further increase in λ0 the fraction of molecules returning onto the surface considerably increases
and the escape flux Ψ sharply decreases, together with the velocity u(r), since in the steady flow Φ = F(r),
where F(r) = 4π r2 n(r)u(r) is the local value of the molecular flux. With further increase in λ0 (curves 6
and 7 in Fig. 1b), M and u continue to decrease on the surface, since Φ decreases with increase in λ0 , while
on this λ0 range n(R0 ) and T (R0 ) are similar in value with n0 and T0 . However, with distance from the
source M increases even more rapidly than for λ0 = 4. This is attributable to the fact that an increase in λ0
leads to a more rapid density decrease with distance from the source and to a change in the velocity increase
mechanism [21]: at λ0 = 10–15 the velocity increases in the same way as in free-molecular outflow, that
is, mainly at the expense of a gradual decrease in the fraction of high-velocity molecules moving toward
the source rather than of an increase in the most probable molecular velocity. It might be expected that at
large values of λ0 the limiting Mach number is also greater than unity; however, the asymptotic parameter
behavior in the outflow into a vacuum is realized at a considerable distance from the source [9, 10] which
increases with λ0 [21] and is considerably greater than the distances considered in this study.
The difference in the structure of the sub- and supercritical flows becomes obvious in comparing the
calculations made for λ0 = 1 and 10 using the HS, PM, and PMLB models (Figs. 2 and 3). At λ0 = 1 < λc1
the limiting gas velocity increases with decrease in Kn0 . The preferential energy transition from the internal
to the translational degrees of freedom of molecules with ascent in the atmosphere provides a greater outflow
velocity of a diatomic gas compared with a monatomic gas. At λ0 = 10 > λc1 the difference in the velocities
of the monatomic and diatomic gases is also considerable but the tendency of the variation in u/C0 with
variation in Kn0 is qualitatively opposite. In this case, a decrease in Kn0 leads to a decrease in u/C0 far
away from the source.
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Fig. 2. Dependence of the gas velocity u/C0 on the distance r/R0 at λ0 = 1 (a) and 10 (b); Kn0 = 1, 0.1, and 0.001
(curves (1–3)); (I) HS model; (II) PM; and (III) PMLB.
Fig. 3. Dependences of the local Mach number M (a, b) and the temperatures of the translational Tt /T0 and internal Ti /T0
degrees of freedom (c, d) on the distance r/R0 : λ0 = 1 (a, c) and 10 (b, d), Kn0 = 0.001; (I–III), (Ia), and (IIIa) relate to M
and Ti /T0 and (IV) to Ti /T0 ; (I) is the HS model, (II) is PM, and (III, IV) are PMLB; (Ia, IIIa) relate to supersonic isentropic
expansion of monatomic and diatomic gases. For curves (Ia, IIIa) the parameters on the sonic surface are obtained by the
DSS method using the PM and PMLB models.
In subcritical outflow the presence of internal degrees of freedom has a considerable effect on the distributions of the Mach number and the temperature of the translational degrees of freedom (Fig. 3). The
influence of the internal degrees of freedom on the Tt /T0 distribution is particularly large (Fig. 3c). The
temperature of the internal degrees of freedom is “frozen” considerably more rapidly than in the case of
supercritical outflow (curves IV in Fig. 3c and 3d) than in the subcritical case due to a rapid decrease in
the density and the collision frequency with the altitude in a gravitationally bound atmosphere at λ0 > λc2 .
However, in both cases there exists a fairly long initial region, where Tt ≈ Ti . A small difference in the Tt /T0
profiles for the monatomic and diatomic gases in supercritical outflow is due to the fact that in this case the
main role is played by heat conduction, while in the flow region under consideration the thermal conductivity coefficients in the HS, PM, and PMLB models differ only slightly due to relatively small temperature
variations.
In Fig. 3 curves Ia and IIIa are obtained using the equations
r2 nu = r∗2 n∗ u∗ ,
T
nγ − 1
=
T∗
,
γ −1
n∗
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T⊥ /T∥
245
T∥ /Ti
Fig. 4. Dependence of the temperature ratios T⊥ /T∥ (a) and T∥ /Ti (b) on r/R0 ; PMLB model, Kn0 = 0.001, λ0 = 0, 1, 2,
2.7, 4, 10, and 15, curves (1–7).
u2
GM
GM
u2
γ k
γ k
T +
−
=
T∗ + ∗ −
.
γ − 1m
2
r
γ − 1m
2
r∗
These calculations correspond to the supersonic isentropic expansion of monatomic (γ = 5/3) and diatomic (γ = 7/5) gases from a sonic surface on which the values of the parameters r∗ , n∗ , u∗ , and T∗ are
taken to be equal to those determined using the DSS method and the PM and PMLB models. Model (2.1)
well describes the structure of the initial supersonic region in subcritical outflow of both monatomic and
diatomic gases.
In the studies of atmosphere dynamics an assertion can be found that in a gravitational field for small λ0
the isentropic outflow of a monatomic gas is impossible [29] and, therefore, the monatomic and diatomic
gas flow structures must be fundamentally different. This assertion is based on the fact that the sphericallysymmetric isentropic flow can be continued into the subsonic region, where dM/dr > 0 at M < 1, only for
γ < 3/2 and 2γ < λ0 < γ /(γ − 1), while for the monatomic gas with γ = 5/3 and the polyatomic gases at
λ0 < 2γ or λ0 > γ /(γ − 1) Eqs. (2.1) have no solutions with dM/dr > 0 at M < 1 [32]. The possibility of
continuing the solution into the subsonic region makes it possible to describe the subsonic flow region using
the isentropic model, since on the source surface the Mach number is usually smaller than unity.
However, the results of the kinetic calculations show that at λ0 < λc1 there is no fundamental difference
in the monatomic and diatomic gas flows. In both cases M(R0 ) < 1 but the initial flow acceleration up to
a transonic velocity occurs in a narrow (at small Kn0 ) region, whose thickness coincides in the order with
the Knudsen layer thickness and where the flow is essentially nonequilibrium and nonisentropic. Thus, the
requirement of the existence of a subsonic isentropic flow region is not important for outflow problems in
a weak gravitational field, since at small λ0 the initial acceleration occurs in the nonequilibrium Knudsen
layer for both monatomic and diatomic gases.
Closer to the upper boundary of the computation domain the subcritical flow differs from an isentropic
flow due to a considerable decrease in the density, an increase in the molecular free path, and intensification
of nonequilibrium effects [9, 10, 13, 17, 21]. On the macroscopic level, the flow nonequilibrium manifests
itself in a difference between T∥ , T⊥ , and Ti , in nonzero limiting values of T∥ and Ti , as r/R0 → ∞, and in
the change in the asymptotic behavior of T⊥ , as compared with both isentropic and free-molecular outflows
[9, 13, 17, 21].
The ratio T⊥ /T∥ characterizing the translational nonequilibrium degree [16, 17] (Fig. 4a) comes closer to
unity throughout the entire computation domain with increase in λ0 at λ0 < λc2 and then decreases sharply at
λ0 > λc2 . In the supercritical outflow the near-surface region, in which T⊥ /T∥ ≈ 1, has a tendency to expand
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with increase in λ0 , whereas far away from the source T⊥ /T∥ at r/R0 = const decreases with increase in
λ0 . In the subcritical outflow of a molecular gas, as r/R0 increases, the thermal energy of the translational
radial motion becomes frozen more rapidly than the energy of the internal degrees of freedom (Fig. 4b). In
the supercritical outflow this tendency is reversed, so that T∥ /Ti decreases with increasing r/R0 . The rapid
decrease in both T⊥ /T∥ and T∥ /Ti at λ0 > λc2 (curves 6 and 7 in Fig. 4) with increase in r/R0 at r/R0 > 2 to 3
is due to a sharp decrease in the gas density at large values of the Jeans parameter, so that the flow turns out
to be near-free-molecular. The Tt /Ti distributions characterizing the translational-rotational nonequilibrium
degree are qualitatively similar with the T⊥ /T∥ distributions presented in Fig. 4a.
As λ0 increases at Kn0 = const, the lower exosphere boundary Rexo comes closer to the source surface
[21]. As a result, for λ0 ≥∼ 12 the energy distribution in the degrees of freedom remains near-equilibrium in
a certain region above Rexo . The expansion of the near-equilibrium flow zone above the exosphere boundary
is probably not connected with the change in the role played by collisions at large λ0 . In a strong gravitational field near the source even in the absence of collisions the velocity distributions of the molecules
moving toward and away from the surface can be only slightly different form one another and from distribution (1.3), since at λ0 ≫ 1 the fraction of escaping molecules is vanishingly small. As a result, in the
free- molecular flow with λ0 ≫ 1 a fairly extended layer of quasistatic atmosphere, in which u ≈ 0 and
T∥ ≈ T⊥ ≈ T0 is formed near the surface; in this layer, the density distribution is governed by the barometric
equation at the source temperature T0 [21].
An analysis of the distribution functions of the radial velocity component of molecules f∥ (r, v′∥ ) =
⟨δ (v∥ − v′∥ )⟩/n(r) (Fig. 5; δ (x) is the Dirac delta function) also confirms the absence of qualitative differences associated with the gas molecule models. At λ0 < λc1 the function f∥ (r, v′∥ ) quantitatively depends
on the gas model. For all the models considered f∥ (r, v′∥ ) is characterized by the fact that the most probable
value of v∥ is similar in value with u. At the same time, at λ0 > λc2 the distribution function depends on ζ
and the molecular collision model only slightly. Qualitatively, f∥ (r, v′∥ ) is characterized by the fact that its
maximum is realized at v∥ = 0 and its position does not change with the altitude in the atmosphere, while
the nonzero gas velocity appears as a result of the difference in the velocity distributions of the molecules
moving toward the source and away from it, which is characteristic of free-molecular outflow.
3. ESCAPE FLUX
√
At λ0 = 0 the relative escape flux Φ/Φ0.0 (Φ0.0 = 4π R20 n0 kT0 /(2π m)) is the flux of the number of
molecules leaving the surface) is a monotonically decreasing function of the Knudsen number; in the hardsphere gas, as Kn0 → 0, it approaches the value of about 0.82 [14, 16, 21] (curve Ia in Fig. 6a). This is
due to the formation of the “return” flux of molecules which, as a result of collisions, return to the source.
The return flux is formed chiefly near the source, where the local escape flux φ (r) rapidly approaches its
asymptotic value Φ (Fig. 7, curves 1). At λ0 = 0 the calculated Φ/Φ0.0 (Kn0 ) dependence is in agreement
with the data [15] (points Ib in Fig. 6a).
At λ0 ≥ 0 the quantity Φ0 = Φ0.0 (1 + λ0 ) exp(−λ0 ), equal to the escape flux from the source surface
in the free-molecular flow with the boundary conditions (1.2)–(1.4), can be taken as the scale for the escape
flux [1]. At λ0 = const ∕= 0 the Φ/Φ0 (Kn0 ) dependence is nonmonotonic and has a maximum at Kn0 =
Knmax (λ0 ) (Fig. 6). The appearance of the maximum is due to the fact that at λ0 > 0, as a result of collisions,
a certain additional fraction of molecules acquires velocities sufficient for escaping from the gravitational
well of a planet. This process of the generation of new molecules moving along hyperbolic trajectories is
efficient far away from the source and leads to a nonmonotonic distribution of the local escape flux φ (r)
which increases with r at r/R0 ≫ 1 (Fig. 7, curves 4 and 5).
The comparative calculations carried out at different R1 /R0 showed that the φ (r) distribution also allows
one to judge of the effect of the outer boundary position R1 /R0 on the flow parameter distributions in the
computation domain. When on the outer boundary the quantity d φ /dr is small, the further increase in R1 /R0
has almost no effect on the flow within the entire computation domain.
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f∥ /C0
v∥ /C0
Fig. 5. Distribution functions f∥ of the radial velocity of molecules v∥ for λ0 = 1 (1) and 10 (2); Kn = 0.001, r/R0 = 20;
(I) monatomic gas, HS model; (II) monatomic
gas, PM model; (III) diatomic gas, PMLB model. Curve (IV) is the
√
Maxwellian distribution at T = T0 : C0 = 2kT0 /m.
Fig. 6. Dependence of the dimensional escape flux Φ/Φ0 on the Knudsen number Kn0 for λ0 = 1 (a) and 10 (b); (I) HS
model; (II) PM; and (III) PMLB; (Ia) HS model at λ0 = 0; (Ib) is obtained in [15] at λ0 = 0 on the basis of the HS model.
At λ0 < λc1 (Fig. 6a) the escape fluxes obtained using different molecule models differ within 10%.
At λ0 > λc2 (Fig. 6b) the difference in the escape fluxes obtained basing on the HS and PM models turns out
to be greater for Kn < Knmax (λ0 ) but it remains within 30% on the Kn0 range considered. At λ0 > λc2 and
Kn0 < Knmax (λ0 ) the escape flux for the diatomic gas (PMLB model) is also 15 to 30% greater than for the
monatomic gas (PM model), since a portion of the internal energy of molecules goes over into their kinetic
energy, which favors an increase in the number of escaping molecules. At λ0 > λc2 and Kn0 < Knmax (λ0 )
the ratio Φ/Φ0 rapidly decreases with decrease in Kn0 .
The degree of the nearness of the numerical solution to the steady-state solution can be estimated from
the distribution of the molecule number flux F (Fig. 7). On the λ0 range considered the quantity F is within
1 to 2% equal to Φ throughout the entire computation domain, except from the surface layer, where for
supercritical λ0 the statistical noise level in F is considerable, since in this layer the gas velocity is small
compared with the thermal speed of molecules and can be hardly estimated with a high accuracy when using
the DSS method. With the distance from the source the statistical noise level in F considerably reduces due
to an increase in M.
The λ0 -dependence of the ratio Φ/Φ0 (Fig. 8) makes it possibly formally to determine the positions of
the boundaries λc1 and λc2 of the transitional range at low Knudsen numbers. In the subcritical outflow Φ ≈
0.82Φ0.0 with the result that with increase in λ0 the ratio Φ/Φ0 increases as ≈ 0.82 exp(λ0 )/(1 + λ0 ). With
further increase in λ0 the ratio Φ/Φ0 rapidly decreases on the transitional range and then again increases in
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F/Φ0 ,
φ /F0
Fig. 7. Dependences of the dimensionless molecule number flux F/Φ0 (I) and the local escape flux φ /Φ0 (II) on the
distance r/R0 ; PMLB model, Kn0 = 0.001, λ0 = 0, 2, 2.7, 4, and 10 (curves (1–5)); F = 4π r2 nu.
Fig. 8. Dimensionless escape flux Φ/Φ0 as a function of the Jeans parameter λ0 for Kn0 = 0.001: (I) HS model; (II) PM;
and (III) PMLB; TI and TIII are the transitional λ0 ranges in the monatomic and diatomic gases [21, 22].
the supercritical outflow. A decrease in Φ/Φ0 on the transitional range is due to a steep decrease in the flow
velocity as the limit of the isentropic flow existence has been attained (cf. curves 4 and 5 in Fig. 1a) [21].
As λc1 and λc2 , the Jeans parameter values corresponding to local maximum and minimum of Φ/Φ0 can
be taken. The calculations of this study make it possible to refine the boundaries of the transitional ranges
determined previously in [21, 22] and to estimate them as λc1 = 2.1 and λc2 = 3 to 4 for the monatomic gas
and λc1 = 2.9 and λc2 = 4 to 5 for the diatomic gas. The transitional ranges for the HS and PM models do
not differ within the error of the calculations.
The displacement of the transitional range toward greater λ0 with increase in the number of the degrees
of freedom of gas molecules can be understood by considering the condition for the existence of supersonic
isentropic outflow, that is, the condition of nonnegativeness of the total enthalpy of the fluid on the sonic
surface: γ kT∗ /(γ − 1)/m + u2∗ /2 − GM/r∗ ≥ 0. In the Kn0 → 0 limit, as r∗ /R0 → 1, this condition is
fulfilled for λ0 ≤ λc , where
γγ +1
T∗∗
λc =
(3.1)
2γ − 1
is the critical value of the Jeans parameter and T∗∗ = T∗ /T0 , as Kn0 → 0. The calculations carried out using
the DSS method at Kn0 = 3 ⋅ 10−4 and λ0 = 0 give T∗∗ = 0.642, which almost coincides with T∗∗ = 0.6434
in [13], and λc = 2.14 for the monatomic gas and T∗∗ = 0.755 and λc = 3.17 for the diatomic gas.
As λ0 increases on the interval λc2 < λ0 <∼ 14, the normalized flux Φ/Φ0 grows, which is due to
an increase in the fraction of the “escaping” molecules generated at the expense of collisions in the upper
atmosphere. At Kn0 = 10−3 the relative difference between the escape fluxes obtained using different models
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of molecules is about 50% for λ0 = 6 and then it decreases to 25% for λ0 = 12 to 15. From the data presented
in Fig. 6b it follows that the relative difference between the escape fluxes for the monatomic and diatomic
gases has a tendency to increase with decrease in Kn0 .
Summary. The kinetic calculations of gas expansion from a spherical source into a vacuum in a gravitational field show that there is no fundamental differences between the monatomic and diatomic gas flows.
In particular, in both cases there exists a narrow transitional range of the Jeans parameter λ0 on which at
low Knudsen numbers the escape flux changes by more than an order. With increase in the number of the
degrees of freedom of molecules the transitional range is displaced toward the greater values of the Jeans
parameter. The quantitative differences in the flow structure for the monatomic and diatomic gases are more
important at comparatively low values of the Jeans parameter which in practice correspond to the atmospheres of cometary nuclei, the objects of the Kuiper belt, and other small celestial bodies. In this case, the
subsonic flow region is completely or partially within a nonequilibrium Knudsen layer. At the same time,
for all the models considered the escape fluxes in the subcritical flow differ within 10%. At λ0 = 6 to 15 the
escape flux for the diatomic gas is 25 to 50% greater than that for the monatomic gas.
The study was carried out with the support of NASA (USA) under grant NNX09AB68G.
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